Theses Honors College 11-2014 A Quasi-Classical Logic for Classical Mathematics Henry Nikogosyan University of Nevada, Las Vegas Follow this and additional works at: https://digitalscholarship.unlv.edu/honors_theses Part of the Logic and Foundations Commons Repository Citation Nikogosyan, Henry, "A Quasi-Classical Logic for Classical Mathematics" (2014). Theses. 21. https://digitalscholarship.unlv.edu/honors_theses/21 This Honors Thesis is protected by copyright and/or related rights. It has been brought to you by Digital Scholarship@UNLV with permission from the rights-holder(s). You are free to use this Honors Thesis in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/or on the work itself. This Honors Thesis has been accepted for inclusion in Theses by an authorized administrator of Digital Scholarship@UNLV. For more information, please contact [email protected]. A QUASI-CLASSICAL LOGIC FOR CLASSICAL MATHEMATICS By Henry Nikogosyan Honors Thesis submitted in partial fulfillment for the designation of Departmental Honors Department of Philosophy Ian Dove James Woodbridge Marta Meana College of Liberal Arts University of Nevada, Las Vegas November, 2014 ABSTRACT Classical mathematics is a form of mathematics that has a large range of application; however, its application has boundaries. In this paper, I show that Sperber and Wilson’s concept of relevance can demarcate classical mathematics’ range of applicability by demarcating classical logic’s range of applicability. Furthermore, I introduce how to systematize Sperber and Wilson’s concept of relevance into a quasi- classical logic that can explain classical logic’s and classical mathematics’ range of applicability. INTRODUCTION Classical mathematics is the most robust and commonly used form of mathematics. It is the kind of mathematics that economist use to calculate a country’s GDP. It is the kind of mathematics manufacturers use to produce cars. It is the kind of mathematics that engineers use to build spaceships. Not only can experts rely on classical mathematics as a deductive tool, but they can also apply it to a greater range of mathematical problems than any competing form of mathematics. As a system, classical mathematics has axioms. Axioms are basic rules that, when combined with rules of inference, govern the way a system operates and reaches conclusions. Classical mathematicians attempt to discover and prove the conclusions that follow from the axioms of classical mathematics. Each system of mathematics uses a system of logic. Classical mathematics uses classical logic. The role of the logic that a form of mathematics uses is to provide some of the basic principles that govern the operation of the mathematics. The logic is also 1 able to explain mathematics by allowing the construction of proofs about objects and properties of mathematics. This makes the relation between a mathematics system and its logic intimate – the logic allows us to see into the mathematics. However, despite the seemingly unshakable nature of classical mathematics its acceptability is questionable. Alternative programs, such as the one developed by the intuitionists, compete with classical mathematics. Some of these competing schools arise from disagreement over which axioms are acceptable. The law of excluded middle (LEM) is one of those controversial axioms. Below is the schema of the law, followed by its traditional interpretation. p v ¬p This schema expresses that the proposition, p, is true or its negation is true. For instance, either Mercury is geologically active or it is not the case that Mercury is geologically active. The structure of LEM applies, or at least is supposed to apply, to any meaningful sentence. Here is a non-traditional interpretation of LEM that includes the schema’s universal application to sentences: for all x (where x can be any sentence), either x is true or it is not the case that x is true. There are differing reasons for rejecting LEM. The constructivists, who do not consider LEM to be an axiom, require a proof of LEM in order for it to be true. However, they cannot derive LEM for all possible cases. The finitude of human experience does not allow the construction of the infinite number of derivations necessary to prove that LEM holds. Others simply do not find the axiom to be self-evident. The restriction to 2 the two options that LEM expresses does not seem to be obvious. The possibility of a third option or infinitely many options remains. Kurt Gödel famously claimed, “the axioms force themselves upon us as being true” (Gödel, 484). Those who question the self-evidence of LEM do not find the dilemma that it expresses to force itself upon us. What I consider the best reasons for rejecting LEM are counterexamples of the following kind. (1) Either an A musical note is green or it is not the case that an A musical note is green. (2) Either the purpose of human life is to be happy or it is not the case that the purpose of human life is to be happy. (3) Either the largest natural number is even or it is not the case that the largest natural number is even. (4) Either there will be a sea battle tomorrow or it is not the case that there will be a sea battle tomorrow. The counterexamples are cases of LEM. If LEM is correct, then the disjunction must be true in all cases. Any case of LEM that is false or appears to be confused qualifies as a counterexample to LEM. 3 Sentences (1), (2), (3), and (4) present cases of LEM where the predicates employed in the embedded sentences have an unusual relation to the subjects of those sentences. In sentence (1), it is not clear how a sound could satisfy (bear the feature of) having or not having a color. The concept of sound and the concept of color have no overlap; the concept of greenness cannot combine with the concept of sound into a complex concept. This makes the attribution of color to sound a category mistake. A category mistake is attributing a property to a thing that cannot possibly have that property. For instance, to claim that this paper is anxious would be a category mistake. Talking about the anxiousness of this paper does not make sense because this paper is inanimate. In a similar manner, a sound cannot possibly have a color. It is nonsensical to associate color and sound together, at least in any way that is stronger than correlation. In other words, color is irrelevant to sound. The result is that the sentence ‘an A musical note is green’ is nonsense or meaningless. To be fair, the defenders of LEM would not accept sentence (1) to be a genuine case of LEM. In order for a sentence to be an instance of LEM, the sentences must be meaningful. Since sentence (1) is not meaningful (by category mistake), sentence (1) is not a real case of LEM. Consequently, this example is not a genuine counterexample. The defenders of LEM are able to cherry pick away these kinds of cases by setting the standard that sentences must be meaningful. This is a fair move. There are good reasons to reject the use of meaningless sentences into the LEM schema. For sentence (2), assume that natural processes are entirely responsible for the rise of humans, and that entities have purpose only if an intelligent being gives purpose to those entities when creating them. The purpose of a kitchen knife, for instance, is to cut. 4 However, a rock does not have purpose because its coming to existence does not include an intelligent being giving it purpose. Humans, like rocks, are a product of an unguided natural process. Humans came to exist without an intelligent being giving them purpose. On this view, human life does not have purpose. However, the disjunction in sentence (2) presupposes that human life has purpose. The problem is that, according to our background assumption, this presupposition is false. Human life does not have purpose (according to the earlier stipulations). The presupposition being false creates a problem for this case of LEM because both disjuncts rely on the presupposition. Each disjunct is attributing features to the purpose of human life, but that attribution would only make sense if human life really had purpose. Since the presupposition is false, the disjunction appears mistaken. The assumption that human life has purpose acts as a foundation for the disjuncts that allows the disjunction to make sense. Without the foundation (because it is a false assumption), there is no warrant to assert the disjunction. Sentence (3) is another instance of LEM that, like sentence (2), has presupposition failure. In this case, the sentence presupposes that there is a largest natural number. Since the natural numbers go on infinitely, there cannot be a largest natural number. Thus, sentence (3) is asserting a predicate about an object that does not exist. The presupposition is false and each disjunct relies on the presupposition. Consequently, the assertion of the disjunction is mistaken. The defenders of LEM may argue that this is a mistaken interpretation of the sentence. In fact, sentence (3) says that either there exists a largest natural number and that number is even or it is not the case that there exists a largest natural number and that number is even. On this interpretation, there is no presupposition. This may be a result of 5 a divergence in intuitions of what the sentence means, but apply this interpretation to the following sentence: ‘Zeus is a god or it is not the case that Zeus is a god’.